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Annotation of 43BSDTahoe/usr.lib/libm/asincos.c, revision 1.1

1.1     ! root        1: /*
        !             2:  * Copyright (c) 1985 Regents of the University of California.
        !             3:  * All rights reserved.
        !             4:  *
        !             5:  * Redistribution and use in source and binary forms are permitted
        !             6:  * provided that the above copyright notice and this paragraph are
        !             7:  * duplicated in all such forms and that any documentation,
        !             8:  * advertising materials, and other materials related to such
        !             9:  * distribution and use acknowledge that the software was developed
        !            10:  * by the University of California, Berkeley.  The name of the
        !            11:  * University may not be used to endorse or promote products derived
        !            12:  * from this software without specific prior written permission.
        !            13:  * THIS SOFTWARE IS PROVIDED ``AS IS'' AND WITHOUT ANY EXPRESS OR
        !            14:  * IMPLIED WARRANTIES, INCLUDING, WITHOUT LIMITATION, THE IMPLIED
        !            15:  * WARRANTIES OF MERCHANTIBILITY AND FITNESS FOR A PARTICULAR PURPOSE.
        !            16:  *
        !            17:  * All recipients should regard themselves as participants in an ongoing
        !            18:  * research project and hence should feel obligated to report their
        !            19:  * experiences (good or bad) with these elementary function codes, using
        !            20:  * the sendbug(8) program, to the authors.
        !            21:  */
        !            22: 
        !            23: #ifndef lint
        !            24: static char sccsid[] = "@(#)asincos.c  5.3 (Berkeley) 6/30/88";
        !            25: #endif /* not lint */
        !            26: 
        !            27: /* ASIN(X)
        !            28:  * RETURNS ARC SINE OF X
        !            29:  * DOUBLE PRECISION (IEEE DOUBLE 53 bits, VAX D FORMAT 56 bits)
        !            30:  * CODED IN C BY K.C. NG, 4/16/85, REVISED ON 6/10/85.
        !            31:  *
        !            32:  * Required system supported functions:
        !            33:  *     copysign(x,y)
        !            34:  *     sqrt(x)
        !            35:  *
        !            36:  * Required kernel function:
        !            37:  *     atan2(y,x) 
        !            38:  *
        !            39:  * Method :                  
        !            40:  *     asin(x) = atan2(x,sqrt(1-x*x)); for better accuracy, 1-x*x is 
        !            41:  *               computed as follows
        !            42:  *                     1-x*x                     if x <  0.5, 
        !            43:  *                     2*(1-|x|)-(1-|x|)*(1-|x|) if x >= 0.5.
        !            44:  *
        !            45:  * Special cases:
        !            46:  *     if x is NaN, return x itself;
        !            47:  *     if |x|>1, return NaN.
        !            48:  *
        !            49:  * Accuracy:
        !            50:  * 1)  If atan2() uses machine PI, then
        !            51:  * 
        !            52:  *     asin(x) returns (PI/pi) * (the exact arc sine of x) nearly rounded;
        !            53:  *     and PI is the exact pi rounded to machine precision (see atan2 for
        !            54:  *      details):
        !            55:  *
        !            56:  *     in decimal:
        !            57:  *             pi = 3.141592653589793 23846264338327 ..... 
        !            58:  *    53 bits   PI = 3.141592653589793 115997963 ..... ,
        !            59:  *    56 bits   PI = 3.141592653589793 227020265 ..... ,  
        !            60:  *
        !            61:  *     in hexadecimal:
        !            62:  *             pi = 3.243F6A8885A308D313198A2E....
        !            63:  *    53 bits   PI = 3.243F6A8885A30  =  2 * 1.921FB54442D18   error=.276ulps
        !            64:  *    56 bits   PI = 3.243F6A8885A308 =  4 * .C90FDAA22168C2    error=.206ulps
        !            65:  *     
        !            66:  *     In a test run with more than 200,000 random arguments on a VAX, the 
        !            67:  *     maximum observed error in ulps (units in the last place) was
        !            68:  *     2.06 ulps.      (comparing against (PI/pi)*(exact asin(x)));
        !            69:  *
        !            70:  * 2)  If atan2() uses true pi, then
        !            71:  *
        !            72:  *     asin(x) returns the exact asin(x) with error below about 2 ulps.
        !            73:  *
        !            74:  *     In a test run with more than 1,024,000 random arguments on a VAX, the 
        !            75:  *     maximum observed error in ulps (units in the last place) was
        !            76:  *      1.99 ulps.
        !            77:  */
        !            78: 
        !            79: double asin(x)
        !            80: double x;
        !            81: {
        !            82:        double s,t,copysign(),atan2(),sqrt(),one=1.0;
        !            83: #if !defined(vax)&&!defined(tahoe)
        !            84:        if(x!=x) return(x);     /* x is NaN */
        !            85: #endif /* !defined(vax)&&!defined(tahoe) */
        !            86:        s=copysign(x,one);
        !            87:        if(s <= 0.5)
        !            88:            return(atan2(x,sqrt(one-x*x)));
        !            89:        else 
        !            90:            { t=one-s; s=t+t; return(atan2(x,sqrt(s-t*t))); }
        !            91: 
        !            92: }
        !            93: 
        !            94: /* ACOS(X)
        !            95:  * RETURNS ARC COS OF X
        !            96:  * DOUBLE PRECISION (IEEE DOUBLE 53 bits, VAX D FORMAT 56 bits)
        !            97:  * CODED IN C BY K.C. NG, 4/16/85, REVISED ON 6/10/85.
        !            98:  *
        !            99:  * Required system supported functions:
        !           100:  *     copysign(x,y)
        !           101:  *     sqrt(x)
        !           102:  *
        !           103:  * Required kernel function:
        !           104:  *     atan2(y,x) 
        !           105:  *
        !           106:  * Method :                  
        !           107:  *                           ________
        !           108:  *                           / 1 - x
        !           109:  *     acos(x) = 2*atan2(  / -------- , 1 ) .
        !           110:  *                        \/   1 + x
        !           111:  *
        !           112:  * Special cases:
        !           113:  *     if x is NaN, return x itself;
        !           114:  *     if |x|>1, return NaN.
        !           115:  *
        !           116:  * Accuracy:
        !           117:  * 1)  If atan2() uses machine PI, then
        !           118:  * 
        !           119:  *     acos(x) returns (PI/pi) * (the exact arc cosine of x) nearly rounded;
        !           120:  *     and PI is the exact pi rounded to machine precision (see atan2 for
        !           121:  *      details):
        !           122:  *
        !           123:  *     in decimal:
        !           124:  *             pi = 3.141592653589793 23846264338327 ..... 
        !           125:  *    53 bits   PI = 3.141592653589793 115997963 ..... ,
        !           126:  *    56 bits   PI = 3.141592653589793 227020265 ..... ,  
        !           127:  *
        !           128:  *     in hexadecimal:
        !           129:  *             pi = 3.243F6A8885A308D313198A2E....
        !           130:  *    53 bits   PI = 3.243F6A8885A30  =  2 * 1.921FB54442D18   error=.276ulps
        !           131:  *    56 bits   PI = 3.243F6A8885A308 =  4 * .C90FDAA22168C2    error=.206ulps
        !           132:  *     
        !           133:  *     In a test run with more than 200,000 random arguments on a VAX, the 
        !           134:  *     maximum observed error in ulps (units in the last place) was
        !           135:  *     2.07 ulps.      (comparing against (PI/pi)*(exact acos(x)));
        !           136:  *
        !           137:  * 2)  If atan2() uses true pi, then
        !           138:  *
        !           139:  *     acos(x) returns the exact acos(x) with error below about 2 ulps.
        !           140:  *
        !           141:  *     In a test run with more than 1,024,000 random arguments on a VAX, the 
        !           142:  *     maximum observed error in ulps (units in the last place) was
        !           143:  *     2.15 ulps.
        !           144:  */
        !           145: 
        !           146: double acos(x)
        !           147: double x;
        !           148: {
        !           149:        double t,copysign(),atan2(),sqrt(),one=1.0;
        !           150: #if !defined(vax)&&!defined(tahoe)
        !           151:        if(x!=x) return(x);
        !           152: #endif /* !defined(vax)&&!defined(tahoe) */
        !           153:        if( x != -1.0)
        !           154:            t=atan2(sqrt((one-x)/(one+x)),one);
        !           155:        else
        !           156:            t=atan2(one,0.0);   /* t = PI/2 */
        !           157:        return(t+t);
        !           158: }

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